Author: Todd_TrimbleFormat: MarkdownItexAdded material to [[injective object]], including a proof of Baer's criterion for injective modules, and the result that for modules over Noetherian rings, direct sums of injective modules are injective.

Added material to injective object, including a proof of Baer’s criterion for injective modules, and the result that for modules over Noetherian rings, direct sums of injective modules are injective.

Author: UrsFormat: MarkdownItexMonths later...
Thanks, Todd! :-)
I have added some hyperlinks and split off the Bass-Papp result as a separate numbered proposition.
I'll now copy (not move) this stuff over to the new entry _[[injective module]]_ (to parallel _[[projective module]]_.)

Months later…

Thanks, Todd! :-)

I have added some hyperlinks and split off the Bass-Papp result as a separate numbered proposition.

Author: UrsFormat: MarkdownItexYears later...
I have made two little lemmas more explicit:
1. right adjoints of left exacts preserves injective objects ([here](https://ncatlab.org/nlab/show/injective+object#RightAdjointsOfExactFunctorsPreserveInjectives));
1. right adjoints of faithful left exacts transfer enough injectives ([here](https://ncatlab.org/nlab/show/injective+object#TransferOfEnoughInjectivesAlongAdjunctions)).
Then I made the use of these two lemmas in the statement that $R Mod$ has enough injectives ([here](https://ncatlab.org/nlab/show/injective+object#RModHasEnoughInjectives)) more explicit.

Author: UrsFormat: MarkdownItexThanks!
By the way, I discovered that we had parts of the lemmas missing at _[[injective object]]_ spelled out at _[[injective module]]_, and vice versa. I have tried to fix that.

Author: IngoBlechschmidtFormat: MarkdownItexFor a right adjoint to preserve injective objects, it suffices for its left adjoint to be merely left exact (instead of exact; with the same proof). I strengthened the formulation of the lemma accordingly.

For a right adjoint to preserve injective objects, it suffices for its left adjoint to be merely left exact (instead of exact; with the same proof). I strengthened the formulation of the lemma accordingly.

Author: IngoBlechschmidtFormat: MarkdownItexOh, of course. Sorry for the noise; the statement now reads: "Given a pair of additive adjoint functors between abelian categories such that the left adjoint L is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects." Also I added a remark that additivity of the left adjoint is given automatically (being exact, the functor preserves biproducts).

Oh, of course. Sorry for the noise; the statement now reads: “Given a pair of additive adjoint functors between abelian categories such that the left adjoint L is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects.” Also I added a remark that additivity of the left adjoint is given automatically (being exact, the functor preserves biproducts).

Author: IngoBlechschmidtFormat: MarkdownItexRight. I think each of the following conditions is sufficient for guaranteeing that a functor $\mathcal{A} \to \mathcal{B}$ preserves biproducts (where $\mathcal{A}$ and $\mathcal{B}$ are categories with a zero object):
1. The functor preserves finite products (for instance, because it's a right adjoint) and any product in $\mathcal{B}$ is a biproduct.
2. The functor preserves finite coproducts (for instance, because it's a left adjoint) and any coproduct in $\mathcal{B}$ is a biproduct.
3. The functor preserves finite products and coproducts.

Right. I think each of the following conditions is sufficient for guaranteeing that a functor &Ascr;&rightarrow;&Bscr;\mathcal{A} \to \mathcal{B} preserves biproducts (where &Ascr;\mathcal{A} and &Bscr;\mathcal{B} are categories with a zero object):

The functor preserves finite products (for instance, because it’s a right adjoint) and any product in &Bscr;\mathcal{B} is a biproduct.

The functor preserves finite coproducts (for instance, because it’s a left adjoint) and any coproduct in &Bscr;\mathcal{B} is a biproduct.

Author: IngoBlechschmidtFormat: MarkdownItexI added to _[[injective object]]_ a couple of observations about internally injective objects in toposes. Somewhat surprisingly (to me), it turns out that external injectivity and internal injectivity actually coincide, in stark contrast to the situation with [[internally projective objects]]. I have checked this only for localic toposes, but believe it's true in more generality; I'll update the entry when I know.

I added to injective object a couple of observations about internally injective objects in toposes. Somewhat surprisingly (to me), it turns out that external injectivity and internal injectivity actually coincide, in stark contrast to the situation with internally projective objects. I have checked this only for localic toposes, but believe it’s true in more generality; I’ll update the entry when I know.

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